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Number of partitions of floor(7n/2)-1 into n nonnegative integers each no greater than 7.
(Formerly M3388 N1368)
2

%I M3388 N1368 #35 Oct 27 2023 21:08:49

%S 0,1,4,10,23,48,94,166,285,464,734,1109,1646,2371,3366,4652,6357,8519,

%T 11309,14754,19103,24399,30956,38797,48355,59665,73264,89145,108011,

%U 129864,155554,185017,219336,258438,303604,354665,413213,479048,554033

%N Number of partitions of floor(7n/2)-1 into n nonnegative integers each no greater than 7.

%C In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(7n/2)-1 involving the letters a, b, c, d, e, f, g, h, having weights 0, 1, 2, 3, 4, 5, 6, 7 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%D A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A001980/b001980.txt">Table of n, a(n) for n = 0..1000</a> (first 86 terms from Vincenzo Librandi)

%H A. Cayley, <a href="http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;q1=second%20memoir%20on%20quantics;rgn=full%20text;cite1=cayley;cite1restrict=author;idno=ABS3153.0002.001;didno=ABS3153.0002.001;view=pdf;seq=00000289">Numerical tables supplementary to second memoir on quantics</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.

%H A. Cayley, <a href="/A001971/a001971.pdf">Numerical tables supplementary to second memoir on quantics</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]

%H <a href="/index/Rec#order_34">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1, 2, -1, -2, 0, -1, 0, 1, 2, 2, -1, -2, 1, -4, 0, 4, -1, 2, 1, -2, -2, -1, 0, 1, 0, 2, 1, -2, 1, -1, -1, 1).

%F Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)(1-x^7z)), where w=floor(7n/2)-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%F G.f.: -(x^24 +3*x^23 +5*x^22 +10*x^21 +17*x^20 +26*x^19 +33*x^18 +45*x^17 +55*x^16 +61*x^15 +63*x^14 +68*x^13 +67*x^12 +68*x^11 +63*x^10 +61*x^9 +55*x^8 +45*x^7 +33*x^6 +26*x^5 +17*x^4 +10*x^3 +5*x^2 +3*x +1)*x / ((x^4+x^3+x^2+x+1) *(x^4-x^2+1) *(x^2+x+1)^2 *(x^2-x +1)^2 *(x^2+1)^3 *(x+1)^5 *(x-1)^7). - _Alois P. Heinz_, Jul 25 2015

%o (PARI) f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)*(1-x^7*z)); n=400; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(7*d/2)-1;print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%Y Cf. A001979.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%E a(0)=0 inserted by _Alois P. Heinz_, Jul 25 2015